Weyl-equivariant splitting conjecture for saturated covers

Let $\overline{G}_0$ be a saturated cover, let $\mathscr{X}_{Q,n}$ be the relevant lattice, and let $\mathscr{X}_{0,Q,n}$ be the Weyl-stable sublattice. Let $W$ act by the usual Weyl action and let $\sigma_{\mathscr{X}_0}$ and $\sigma_{\mathscr{X}}$ be the associated permutation representations. Weyl-equivariant splitting conjecture. There exists a $W$-equivariant splitting $s:\mathscr{X}_{Q,n}^\Gamma\to\mathscr{X}_{Q,n}$. Consequently, for every $z\in s(\mathscr{X}_{Q,n}^\Gamma)$, $\sigma_{\mathscr{X}_0,z}\simeq\sigma_{\mathscr{X}_0}$ and $\sigma_{\mathscr{X}}=|\mathscr{X}_{Q,n}^\Gamma|\cdot\sigma_{\mathscr{X}_0}$; in particular, the corresponding Whittaker dimensions satisfy

$$\dim\operatorname{Wh}_\psi(\pi_{\overline{\overline{\overline{\overline{\chi}}}},S})=|\mathscr{X}_{Q,n}^\Gamma|\cdot\dim\operatorname{Wh}_\psi(\pi_{\overline{\overline{\overline{\overline{\chi}}}}_0,S}).$$

This conjecture would make the Weyl action compatible with the lattice splitting and yield the stated multiplicity formula for Whittaker models.